Analysis of Algorithms

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A quite general model of source that comes from dynamical systems theory is introduced. Within this model, basic problems of algorithmic information theory contexts are analysed. The main tool for the analysis is a (generalized) transfer operator, which can be viewed as a "generating" operator for "fundamental" probabilities. Its dominant spectral objects are related to important parameters of the source, such as the entropy rate, and play a central rôle in all the results.

The traditional zero-one principle for sorting networks states that "if a network with n input lines sorts all 2 n binary sequences into nondecreasing order, then it will sort any arbitrary sequence of n numbers into nondecreasing order". We generalize this to the situation when a network sorts almost all binary sequences and relate it to the behavior of the sorting network on arbitrary inputs. We also present an application to mesh sorting.

A new algorithm for generating order preserving minimal perfect hash functions is presented. The algorithm is probabilistic, involving generation of random graphs. It uses expected linear time and requires a linear number of words to represent the hash function, and thus is optimal up to constant factors. It runs very fast in practice.

Previously, it was shown in a paper by Kaldewaij and Schoenmakers that for topdown skew heaps the amortized number of comparisons required for meld and delmin is upper bounded by log φ n, where n is the total size of the inputs to these operations and φ = ( √ 5 + 1)/2 denotes the golden ratio. In this paper we present worst-case sequences of operations on top-down skew heaps in which each application of meld and delmin requires approximately log φ n comparisons. As the remaining heap operations require no comparisons, it then follows that the set of bounds is tight. The result relies on a particular class of self-recreating binary trees, which is related to a sequence known as Hofstadter's G-sequence.

The increasing use of phylogeny in biological studies is limited by the need to make available more efficient tools for computing distances between trees. The geodesic tree distance-introduced by Billera, Holmes, and Vogtmann-combines both the tree topology and edge lengths into a single metric. Despite the conceptual simplicity of the geodesic tree distance, algorithms to compute it don't scale well to large, real-world phylogenetic trees composed of hundred or even thousand leaves. In this paper, we propose the geodesic distance as an effective tool for exploring the likelihood profile in the space of phylogenetic trees, and we give a cubic time algorithm, GEOHEURISTIC, in order to compute an approximation of the distance. We compare it with the GTP algorithm, which calculates the exact distance, and the cone path length, which is another approximation, showing that GEOHEURISTIC achieves a quite good trade-off between accuracy (relative error always lower than 0.0001) and efficiency. We also prove the equivalence among GEOHEURISTIC, cone path, and Robinson-Foulds distances when assuming branch lengths equal to unity and we show empirically that, under this restriction, these distances are almost always equal to the actual geodesic.

We present short proofs on the mistake bounds of the 1-nearest neighbor algorithm on an online prediction problem of path labels. The algorithm is one of key ingredients in the algorithm by Herbster, Lever, and Pontil for general graphs. Our proofs are combinatorial and naturally show that the algorithm works when the set of labels is not binary.

Data sorting is one of the most intensively studied problems in computing science for both its theoretical importance and its use in many applications. Quicksort which depends on an appropriate pivot selection technique for its performance is widely considered to be one of the most efficient sorting techniques. Brest et al. (2000) has implemented a parallel quicksort algorithm on PC-cluster using a Median5 function as a pivot selection scheme. In this paper, a sequential quicksort was implemented using Median5 function as a pivot selection scheme and subsequently a new pivot selection scheme for minimizing the execution time of quicksort algorithm sequentially is proposed. The two schemes were tested together using integer and double array data types. From the results obtained, the execution time of quicksort algorithm was reduced by about 23-28% for integer array and 17-22% for double array when compared with Median5 function (median-of-five with random index selection scheme).

Wireless sensor networks (WSNs) are emerging as an effective means for environment monitoring. This paper investigates a strategy for energy efficient monitoring in WSNs that partitions the sensors into covers, and then activates the covers iteratively in a round-robin fashion. This approach takes advantage of the overlap created when many sensors monitor a single area. Our work builds upon previous work in , where the model is first formulated. We have designed three approximation algorithms for a variation of the SET K-COVER problem, where the objective is to partition the sensors into covers such that the number of covers that include an area, summed over all areas, is maximized. The first algorithm is randomized and partitions the sensors, in expectation, within a fraction 1 -1 e (∼.63) of the optimum. We present two other deterministic approximation algorithms. One is a distributed greedy algorithm with a 1 2 approximation ratio and the other is a centralized greedy algorithm with a 1-1 e approximation ratio. We show that it is NP-Complete to guarantee better than 15 16 of the optimal coverage, indicating that all three algorithms perform well with respect to the best approximation algorithm possible in polynomial time, assuming P = N P . Simulations indicate that in practice, the deterministic algorithms perform far above their worst case bounds, consistently covering more than 72% of what is covered by an optimum solution. Simulations also indicate that the increase in longevity is proportional to the amount of overlap amongst the sensors. The algorithms are fast, easy to use, and according to simulations, significantly increase the longevity of sensor networks. The randomized algorithm in particular seems quite practical.

Breadth First Search (BFS) is a building block for graph algorithms and has recently been used for large scale analysis of information in a variety of applications including social networks, graph databases and web searching. Due to its importance, a number of different parallel programming models and architectures have been exploited to optimize the BFS. However, due to the irregular memory access patterns and the unstructured nature of the large graphs, its efficient parallelization is a challenge. The Xeon Phi is a massively parallel architecture available as an off-the-shelf accelerator, which includes a powerful 512 bit vector unit with optimized scatter and gather functions. Given its potential benefits, work related to graph traversing on this architecture is an active area of research. We present a set of experiments in which we explore architectural features of the Xeon Phi and how best to exploit them in a top-down BFS algorithm but the techniques can be applied to the current state-of-the-art hybrid, top-down plus bottom-up, algorithms. We focus on the exploitation of the vector unit by developing an improved highly vectorized OpenMP parallel algorithm, using vector intrinsics, and understanding the use of data alignment and prefetching. In addition, we investigate the impact of hyperthreading and thread affinity on performance, a topic that appears under researched in the literature. As a result, we achieve what we believe is the fastest published top-down BFS algorithm on the version of Xeon Phi used in our experiments. The vectorized BFS topdown source code presented in this paper can be available on request as free-to-use software.

Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of -1 phases ( is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the average-case and the distributional analysis, and by handling in parallel the integer and the polynomial cases, for polynomials with coefficients in a finite field. The main parameters under consideration are the number of iterations in each phase and the evolution of the size of the current gcd along the execution. Three phenomena are clearly emphasized through this analysis: the fact that almost all the computations are performed during the first phase, the great difference between the probabilistic behavior of the first phase compared to subsequent phases, and, as can be expected, the great similarity between the integer and the polynomial cases.

There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity µ(n) and stop the recursion at a depth slightly smaller than lg n. A rough estimate of the worst-case complexity of these fast versions provides the bound O(µ(n) log n). Even the worst-case estimate is partly based on heuristics and is not actually proven. Here, we provide a precise probabilistic analysis of some of these fast variants, and we prove that their average bit-complexity on random inputs of size n is Θ(µ(n) log n, with a precise remainder term, and estimates of the constant in the Θ-term. Our analysis applies to any cases when the cost µ(n) is of order Ω(n log n), and is valid both for the FFT multiplication algorithm of Schönhage-Stassen, but also for the new algorithm introduced quite recently by Fürer [12]. We view such a fast algorithm as a sequence of what we call interrupted algorithms, and we obtain two main results about the (plain) Euclid Algorithm which are of independent interest. We precisely describe the evolution of the distribution of numbers during the execution of the (plain) Euclid Algorithm, and we exhibit an (unexpected) density ψ which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the interrupted algorithms are locally "similar" to the total algorithm. This finally leads to the precise evaluation of the average bit-complexity of these fast algorithms. This work uses various tools, and is based on a precise study of generalised transfer operators related to the dynamical system underlying the Euclid Algorithm.

We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bit-complexity which involves two main parameters : digit-costs and length of remainders. We first show here that an asymptotic gaussian law holds for the length of remainders at a fraction of the execution, which exhibits a deep regularity phenomenon. Then, we study in each framework-polynomials (P) and integer numbers (I)-two gcd algorithms, the standard one (S) which only computes the gcd, and the extended one (E) which also computes the Bezout pair, and is widely used for computing modular inverses. The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the Extended algorithm: we exhibit an asymptotic gaussian law for the bit-complexity of the extended algorithm, in both cases (P) and (I). We also prove that an asymptotic gaussian law for the bit-complexity of the standard gcd in case (P), but we do not succeed obtaining a similar result in case (I). The integer study is more involved than the polynomial study, as it is usually the case. In the polynomial case, we deal with the central tools of the distributional analysis of algorithms, namely bivariate generating functions. In the integer case, we are led to dynamical methods, which heavily use the dynamical system underlying the number Euclidean algorithm, and its transfer operator. Baladi and Vallée [2] have recently designed a general framework for "distributional dynamical analysis", where they have exhibited asymptotic gaussian laws for a large family of parameters. However, this family does not contain neither the bit-complexity cost nor the size of remainders, and we have to extend their methods for obtaining our results. Even if these dynamical methods are not necessary in case (P), we explain how the polynomial dynamical system can be also used for proving our results. This provides a common framework for both analyses, which well explains the similarities and the differences between the two cases (P) and (I), for the algorithms themselves, and also for their analysis. An extended abstract of this paper can be found in Proceedings of LATIN'06 [21].

Andersson and Nilsson have already shown that the average depth D n of random LC-tries is only Θ (log * n) when the keys are produced by a symmetric memoryless process, and that D n = O (log log n) when the process is asymmetric. In this paper we refine the second estimate by showing that asymptotically (with n → ∞): Dn ∼ 1 η log log n, where n is the number of keys inserted in a trie, η = -log (1 -h/h -∞ ), h = -p log p -q log q is the entropy of a binary memoryless source with probabilities p, q = 1 -p (p = q), and h -∞ = -log min(p, q).

We consider policies that manage fixed-size dynamic linear lists, when the references follow the independent reference model. We define the counter scheme. a policy that keeps lhe records sorted by their access frequencies, and prove that among all detenninistic policies it produces the least expected cost of access, at any time.

The field of parallel metaheuristics is continuously evolving as a result of new technologies and needs that researchers have been encountering. In the last decade, new models of algorithms, new hardware for parallel execution/communication, and new challenges in solving complex problems have been making advances in a fast manner. We aim to discuss here on the state of the art, in a summarized manner, to provide a solution to deal with some of the growing topics. These topics include the utilization of classic parallel models in recent platforms (such as grid/cloud architectures and GPU/APU). However, porting existing algorithms to new hardware is not enough as a scientific goal, therefore researchers are looking for new parallel optimization and learning models that are targeted to these new architectures. Also, parallel metaheuristics, such as dynamic optimization and multiobjective problem resolution, have been applied to solve new problem domains in past years. In this article, ...

Buffer management for a D-disk parallel I/O system is considered in the context of randomized placement of data on the disks. A simple prefetching and caching algorithm PHASE-LRU using bounded lookahead is described and analyzed. It is shown that PHASE-LRU performs an expected number of I/Os that is within a factor (log D/ log log D) of the number performed by an optimal off-line algorithm. In contrast, any deterministic buffer management algorithm with the same amount of lookahead must do at least (√ D) times the number of I/Os of the optimal.

Lafayette, IN, a position he has held since 1976. His research interests include algorithms for data storage and retrieval, programming languages, and software engineering. Dr. Comer is a member of the Association for Computing Machinery and Sigma Xi.

A graph G = (V, E) is distance hereditary if every induced path of G is a shortest path. In this paper, we show that the eccentricity function e(v) = max{d(v, u) : u ∈ V } in any distance-hereditary graph G is almost unimodal, that is, every vertex v with e(v) > rad(G) + 1 has a neighbor with smaller eccentricity. Here, rad(G) = min{e(v) : v ∈ V } is the radius of graph G. Moreover, we use this result to fully characterize the centers of distance-hereditary graphs. Several bounds on the eccentricity of a vertex with respect to its distance to the center of G or to the ends of a diametral path are established. Finally, we propose a new linear time algorithm to compute all eccentricities in a distance-hereditary graph.

We consider the problem of online robot navigation in an unfamiliar two-dimensional environment, using comparatively limited sensing information. In particular, the robot has local sensors to detect the proximity of obstacles and permit boundary-following, and is able to determine its current distance and relative bearing to its nal destination (via distance queries). By contrast, most previous algorithms for online navigation have assumed that the robot knows its exact current position. Because determining exact location is prone to error that accumulates over time, the usefulness of such algorithms may be limited. In contrast, distance queries give less information, but the accuracy of each query is independent of the number of queries, which means distance queries can be more robust. We formally de ne our model and give new, e cient navigation algorithms and lower bounds for this setting.

The generalized Tower of Hanoi problem with h ≥ 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h, n and the source and destination pegs.

Hamming graphs are, by definition, the Cartesian product of complete graphs. In the bipartite case these graphs are hypercubes. We present an algorithm recognizing Hamming graphs in linear time and space. This improves a previous algorithm which was linear in time but not in space. This also favorably compares to the general decomposition algorithms of graphs with respect to the Cartesian product, none of which is linear.

We start with a set of n players. With some probability P (n, k), we kill n − k players; the other ones stay alive, and we repeat with them. What is the distribution of the number X n of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions P (n, k), including stochastic monotonicity and the assumption that roughly a fixed proportion α of the players survive in each round. We prove a kind of convergence in distribution for X n − log 1/α n; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable Z such that d(X n , ⌈Z +log 1/α n⌉) → 0, where d is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin [7]. We study the latter algorithm further, including numerical results.

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Teaching analysis of algorithms to students in Computer Science degrees, using the approach popularized by Knuth in his series of books ``The Art of Computer Programming" and later by Sedgewick and Flajolet in the book ``An Introduction to the Analysis of Algorithms", is not a simple task since, in general, these students are more interested in the implementation of an algorithm than in the corresponding theoretical aspects. This approach concentrates on precisely characterizing the performance of algorithms by determining their best, worst and average case performance using a methodology based on symbolic tools such as recurrence relations and generating functions.The most difficult aspect is to understand the average case since this corresponds to studying the algorithm as its possible inputs vary: this represents the most important goal since generally students have no difficulty in understanding the best and worst cases, corresponding to particular input configurations...

We report on ongoing experimental work on evaluating test sets for testing modal satisfiability solvers. Our longterm aim is to understand the difference between the use of structured and of unstructured randomly generated problems. Which parts of the problem space do they explore? * Juan Heguiabehere and Maarten de Rijke were supported by the Spinoza project 'Logic in Action.' Gabriel Infante López and Maarten de Rijke were supported by a grant from the Netherlands Organization for Scientific Research (NWO) under project number 220-80-001. Maarten de Rijke was also supported by grants from NWO under project numbers 612-13-001, 365-20-005, 612.069.006, and 612.000.106. We would like to thank our anonymous referees for valuable comments.

Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a Pcomplete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in a time much less than the depth of the circuit. Nonetheless, it is found that, for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circuit evaluation is apparently hard. In particular, for some random circuits composed of connectives with five or more inputs, the number of true outputs at each level is a chaotic sequence. Finally, while the average case complexity depends on the choice of ensemble, it is shown that for all ensembles it is possible to simultaneously construct a typical circuit together with its solution in polylogarithmic parallel time.

A connected ordering (v1, v2,. .. , vn) of V (G) is an ordering of the vertices such that vi has at least one neighbour in {v1,. .. , vi−1} for every i ∈ {2,. .. , n}. A connected greedy coloring (CGC for short) is a coloring obtained by applying the greedy algorithm to a connected ordering. This has been first introduced in 1989 by Hertz and de Werra, but still very little is known about this problem. An interesting aspect is that, contrary to the traditional greedy coloring, it is not always true that a graph has a connected ordering that produces an optimal coloring; this motivates the definition of the connected chromatic number of G, which is the smallest value χc(G) such that there exists a CGC of G with χc(G) colors. An even more interesting fact is that χc(G) ≤ χ(G) + 1 for every graph G (Benevides et. al. 2014). In this paper, in the light of the dichotomy for the coloring problem restricted to H-free graphs given by Král et.al. in 2001, we are interested in investigating the problems of, given an H-free graph G: (1). deciding whether χc(G) = χ(G); and (2). given also a positive integer k, deciding whether χc(G) ≤ k. We have proved that Problem (2) has the same dichotomy as the coloring problem (i.e., it is polynomial when H is an induced subgraph of P4 or of P3 + K1, and it is NPcomplete otherwise). As for Problem (1), we have proved that χc(G) = χ(G) always hold when G is an induced subgraph of P5 or of P4 + K1, and that it is NP-complete to decide whether χc(G) = χ(G) when H is not a linear forest or contains an induced P9. We mention that some of the results actually involve fixed k and fixed χ(G).

Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a Pcomplete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in a time much less than the depth of the circuit. Nonetheless, it is found that, for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circuit evaluation is apparently hard. In particular, for some random circuits composed of connectives with five or more inputs, the number of true outputs at each level is a chaotic sequence. Finally, while the average case complexity depends on the choice of ensemble, it is shown that for all ensembles it is possible to simultaneously construct a typical circuit together with its solution in polylogarithmic parallel time.

The generalized Tower of Hanoi problem with h ≥ 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h, n and the source and destination pegs.

Generalized dimension exchange (GDE) has been proposed as a method of load balancing in interconnection networks. This paper examines the behavior of GDE over the family of binary Hamming graphs. A lower bound for the rate at which GDE converges to load balance is also given.

Pairwise comparison (PC) is a well‐established method to assist decision makers (DMs) in estimating their preferences. This paper considers the rationale, design, and evaluation of an open‐source priority estimation tool, PriEsT, which has been developed to offer new features related to the PC method. PriEsT is able to assist DMs in interactively identifying and revising their judgments based on different consistency measures and graphical aids. When inconsistency cannot be improved due to practical limitations, PriEsT offers a wide range of Pareto‐optimal solutions based on multiobjective optimization, unlike other tools that offer only a single solution. DMs have the flexibility to select any of these nondominated solutions according to their requirements. The features of PriEsT have been demonstrated and evaluated through its application to a real‐world case study: the selection of the most appropriate telecom infrastructure for rural areas. This case study using PriEsT has highl...

We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures.Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely related w̄(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = Σaw(t, a), i.e. as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel–Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too.In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries.The final section of this report briefly summarizes and i...

In this paper we consider the skip graph data structure, a load balancing alternative to skip lists, designed to perform better in a distributed environment. We extend previous results of Devroye on skip lists, and prove that the maximum length of a search path in a random binary skip graph of size n is of order log n with high probability.

Random Noise and Random Walk algorithms are local search strategies that have been used for the problem of satisfiability testing (SAT). We present a Markov-chain based analysis of the performance of these algorithms. The performance measures we consider are the probability of finding a satisfying assignment and the distribution of the best solution observed on a given SAT instance. The analysis provides exact statistics, but is restricted to small problems as it requires the storage and use of knowledge about the entire search space. We examine the effect of p, the probability of making non-greedy moves, on these algorithms and provide a justification for the practice of choosing this value empirically.

We present the convergence analysis of convex combination of the alternating projection and Douglas–Rachford operators for solving the phase retrieval problem. New convergence criteria for iterations generated by the algorithm are established by applying various schemes of numerical analysis and exploring both physical and mathematical characteristics of the phase retrieval problem. Numerical results demonstrate the advantages of the algorithm over the other widely known projection methods in practically relevant simulations.

We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity—or inverse calmness—of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural by-product of the framework. To demonstrate the application of the theory, we prove, for the first time, a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, stro...

We consider a single-machine scheduling problem with equal-sized jobs. The objective is to minimize the maximum weighted earliness-tardiness and due-date costs. We present an algorithm to solve this problem. Our algorithm makes use of bottleneck jobs and priority queues, and has a computational complexity of O(n 4 log n). This complexity is a significant improvement of the existing algorithm in the literature.

We study the problem of bicoloring random hypergraphs, both numerically and analytically. We apply the zero-temperature cavity method to find analytical results for the phase transitions (dynamic and static) in the 1RSB approximation. These points appear to be in agreement with the results of the numerical algorithm. In the second part, we implement and test the Survey Propagation algorithm for specific bicoloring instances in the so called HARD-SAT phase.

However difficult the fundamental problems of theoretical computer science may seem, there is very little to suggest that they are anything more than knotty combinatorial problems. So, when we look for reasons for our inability to resolve P = NP and related questions, we most likely find them dealing with a lack of understanding of particular computational problems and their lower bounds. This is the sense of Hopcroft's prediction: "...within the next five years, nobody will prove that ~ny of these problems takes more than let's say n ~ time. I think that's a reasonably safe conjecture and it also illustrates how little we know about lower bounds." [MT]. Hopcroft's guess is uncanny in its accuracy-after six years and considerable effort by many researchers, his conjecture remains unchallenged. The results in this paper offer a possible explanation for our failure to resolve these problems. Roughly, the main result of the sequel links lower bounds and a branch of mathematical logic known as model theory. In particular, we prove that the existence of nonpolynomial lower bounds is equivalent to the existence of nonstandard models of a sizable fragment of arithmetic. Since these are deep logical issues and there are very few techniques for handling them, and since the nonstandard models in question are noneffective, it seems plausible that this linking of complexity theory and logic explains our failure to obtain nontrivial lower bounds. One of the aims of mathematical logic is to clarify the relation between mathematical theories and their interpretations-or models. In logic, a theory is simply a collection of statements and all of their logical consequences, that is, a